Probability distributions of SIR in a Rayleigh fading channel

In fading channel when the interfere signal is a random variable, the resulting signal-to-interference-plus-noise ratio (SINR) or signal-to-interference ratio (SIR) is also random and its distribution has to be known for a correct description of the receiver. In this paper, we derive in closed form expression the probability distributions of SIR when more than one Rayleigh distributed interfere signals are present. SINR or SIR involves the sum and ratio of random variables. When the interfere signals are more than one, it becomes very difficult, if not impossible, to obtain a tractable closed form expression of the probability density function (PDF) or cumulative density function (CDF) of SINR or SIR. The SIR is studied in noise limited systems, i.e. where the interfere signals dominate over noise. If the noise is to be accounted for, which is typically AWGN, the SINR may be studied. In our case, adding AWGN has no effect on the overall probability distribution except for a change in its moments. So to keep the mathematical complexity at bay, only the SIR is considered.